I. KNOTS, LINKS, AND INVARIANT POLYNOMIALS

INTRODUCTION

Basic Definitions

REIDEMEISTER MOVES. KNOT ARITHMETICS

Polygonal Links and Reidemeister Moves

Knot Arithmetics and Seifert Surfaces

LINKS IN 2 SURFACES IN R^3. SIMPLEST LINK INVARIANTS

Knots in Surfaces. The Classiffcation of Torus Knots

The Linking Coefficient

The Arf Invariant

The Colouring Invariant

FUNDAMENTAL GROUP. THE KNOT GROUP

Digression. Examples of Unknotting

Fundamental Group. Basic Definitions and Examples

Calculating Knot Groups

THE KNOT QUANDLE AND THE CONWAY ALGEBRA

Introduction

Geometric and Algebraic Definitions of the Quandle

Completeness of the Quandle

Special Realisations of the Quandle: Colouring Invariant, Fundamental Group, Alexander Polynomial

The Conway Algebra and Polynomial Invariants

Realisations of the Conway Algebra. The Conway-Alexander, Jones, HOMFLY and Kauffman Polynomials

More on Alexander's polynomial. Matrix representation

KAUFFMAN'S APPROACH TO JONES POLYNOMIAL

State models in Physics and Kauffman's Bracket

Kauffman's Form of Jones Polynomial and Skein Relations

Kauffman's Two-Variable Polynomial

PROPERTIES OF JONES POLYNOMIALS. KHOVANOV'S COMPLEX

Simplest Properties

Tait's First Conjecture and Kauffman-Murasugi's Theorem

Menasco-Thistletwaite Theorem and the Classification of Alternating Links

The Third Tait Conjecture

A Knot Table

Khovanov's Categorification of the Jones Polynomial

The Two Phenomenological Conjectures

II. THEORY OF BRAIDS

Braids, Links and Representations of Braid Groups

Four Definitions of the Braid Group

Links as Braid Closures

Braids and the Jones Polynomial

Representations of the Braid Groups

The Krammer-Bigelow Representation

BRAIDS AND LINKS. BRAID CONSTRUCTION ALGORITHMS

Alexander's Theorem

Vogel's Algorithm

ALGORITHMS OF BRAID RECOGNITION

The Curve Algorithm for Braid Recognition

LD-Systems and the Dehornoy Algorithm

Minimal Word Problem for Br(3)

Spherical, Cylindrical, and other Braids

MARKOV'S THEOREM. THE YANG-BAXTER EQUATION

Markov's Theorem after MORTON

Makanin's Generalisations. Unary Braids

Yang-Baxter Equation, Braid Groups and Link Invariants

III. VASSILIEV'S INVARIANTS

Definition and Basic Notions of Vassiliev Invariant Theory

Singular Knots and the Definition of Finite-Type Invariants

Invariants of Orders Zero and One

Examples of Higher-Order Invariants

Symbols of Vassiliev's Invariants Coming from the Conway Polynomial

Other Polynomials and Vassiliev's Invariants

An Example of an Infinite-Order Invariant

THE CHORD DIAGRAM ALGEBRA

Basic Structures

Bialgebra Structure of Algebras A^c and A^t. Chord Diagrams and Feynman diagrams

Lie Algebra Representations, Chord Diagrams, and the Four Colour Theorem

Dimension estimates for Ad. A Table of Known Dimensions

THE KONTSEVICH INTEGRAL AND FORMULAE FOR THE VASSILIEV INVARIANTS209

Preliminary Kontsevich Integral

Z(8) and the Normalisation

Coproduct for Feynman Diagrams

Invariance of the Kontsevich Integral

Vassiliev's Module

Goussarov's Theorem

IV. ATOMS AND d-DIAGRAMS

ATOMS, HEIGHT ATOMS AND KNOTS

Atoms and Height Atoms

Theorem on Atoms and Knots

Encoding of Knots by d-diagrams

d-Diagrams and Chord Diagrams. Embeddability Criterion

A New Proof of the Kauffman-Murasugi Theorem

THE BRACKET SEMIGROUP OF KNOTS

Representation of Long Links by Words in a Finite Alphabet

Representation of Links by Quasitoric Braids

V. VIRTUAL KNOTS

BASIC DEFINITIONS AND MOTIVATION

Combinatorial Definition

Projections from Handle Bodies

Gauss Diagram Approach

Virtual Knots and Links and their Simplest Invariants

Invariants Coming from the Virtual Quandle

INVARIANT POLYNOMIALS OF VIRTUAL LINKS

The Virtual Grouppoid (Quandle)

The Jones-Kauffman Polynomial

Presentations of the Quandle

The V A-Polynomial

Properties of the V A-Polynomial

Multiplicative Approach

The Two-Variable Polynomial

The Multivariable Polynomial

GENERALISED JONES-KAUFFMAN POLYNOMIAL

Introduction. Basic Definitions

An Example

Atoms and Virtual Knots. Minimality Problems

LONG VIRTUAL KNOTS AND THEIR INVARIANTS

Introduction

The Long Quandle

Colouring Invariant

The V-Rational Function

Virtual Knots versus Long Virtual Knots

VIRTUAL BRAIDS

Definitions of Virtual Braids

Burau Representation and its Generalisations

Invariants of Virtual Braids

Virtual Links as Closures of Virtual Braids

An Analogue of Markov's Theorem

VI. OTHER THEORIES

3-MANIFOLDS AND KNOTS IN 3-MANIFOLDS

Knots in RP^3

An Introduction to the Kirby Theory

The Witten Invariants

Invariants of Links in Three-Manifolds

Virtual 3-Manifolds and their Invariants

LEGENDRIAN KNOTS AND THEIR INVARIANTS

Legendrian Manifolds and Legendrian Curves

Definition, Basic Notions, and Theorems

Fuchs-Tabachnikov Moves

Maslov and Bennequin Numbers

Finite-type Invariants of Legendrian Knots

The Differential Graded Algebra (DGA) of a Legendrian Knot

Chekanov-Pushkar' Invariants

Basic Examples

APPENDICES

Independence of Reidemeister Moves

Vassiliev's Invariants for Virtual Links

Energy of a Knot

Unsolved Problems in Knot Theory

A Knot Table

BIBLIOGRAPHY

INDEX